Beat together butter and sugar in a large bowl with an electric mixer until creamy. Beat in egg until well combined.

Whisk together flour, cornmeal, baking powder, and salt in a medium bowl.

Gradually add flour mixture to butter mixture; stir until incorporated. Beat vanilla into butter mixture.

Form dough into a ball, wrap tightly in plastic wrap, and refrigerate until firm, at least 1 hour.

Preheat the oven to 350°F (175°C). Lightly grease baking sheets.

Roll dough out onto a lightly floured surface to 1/8-inch thick. Cut dough with a round or Linzer cookie cutter. Use a smaller cutter to cut the center from the tops.

Place cookie bottoms and tops 1-inch apart on the prepared baking sheets.

Bake in the preheated oven until edges are lightly golden, 10 to 12 minutes. Transfer to wire racks to cool completely.

To make filling, stir together cranberries, brown sugar, and water in a small saucepan over medium-high heat. Cook until cranberries are soft, about 10 minutes. Stir butter and lemon juice into cranberry mixture; remove from heat and cool.